What is and how to use a Fake GI Dome?
It's been a very long time since I read about fake GI but as far as I can remember, Fake GI and light domes are two different approaches to get global illumination. 

The goal of both methods is to create sort of a "50% grey" effect that illuminates the whole scene in a similar way cycles does when you render out the default cube with no additional light source in the scene. Ideally there would be no shadows at all since the light comes from all directions. Additional lights will then produce shadows which aren't completely black. 

afaik Fake GI would use a skydome with maybe a fancy HDRI image slapped on that emits the light, while light domes are dome-like structures build out of actual light sources - many light sources.  

I don't know if those methods are really that much faster to justify the amount of pita they require in terms of workflow (compared to modern render methods, that is), but most of the results I've seen are quality-wise somewhere in between Blender internal with Hemi + additional light sources and Cycles. The point is that shadows are still "harder" thus more unrealistic.